1. PH 401 Dr. Cecilia Vogel

2. Review Outline  Time dependent perturbations  integrating out the time  oscillatory perturbation  stimulated emission (laSEr)  Time dependent perturbations  approximations  perturbation symmetry

3. Time-dependent Perturbation  Recall last time we derived the following eqn for evolution of the amplitude for a state |  f >  given that the particle started in state |  i >  and that a 1 st order approximation was in order  c’ f (t) =(-i/  ) e i(E f -E i )t/   The derivative = computable ftn of t, that can in principle be integrated to get c f (t).

4. Oscillatory Perturbation  Let‘s suppose that the perturbation is a sinusoidal function of time  V pert =V o cos(  pert t)  This would be the case if your particle were being perturbed by the oscillating E-field of a light wave, for example.  c’ f (t) =(-i/  ) e i(E f -E i )t/   c’ f (t) =(-i/  ) e i(E f -E i )t/  cos(  pert t)  The ftn of t is clear in this case, and can be integrated to get c f (t).

5. Oscillatory Perturbation  c’ f (t) =(-i/  ) e i(E f -E i )t/  cos(  pert t)  c f (t) =(-i/  ) ∫e i(E f -E i )t/  cos(  pert t)dt  integral from some initial time, often taken to be t=0, to some final time, tf.  Let A= (-i/  ) (time indep)  c f (t) =(A/2) ∫e i(E f -E i )t/  (e i  pert t + e -i  pert t )dt  c f (t) =(A/2) ∫(e i[ (E f -E i )/  +i  pert ]t + e i[ (E f -E i )/  -i  pert ]t )dt  do the integral….

6. Oscillatory Perturbation  c f (t) =(A/2) ∫(e i[ (E f -E i )/  +  pert ]t + e i[ (E f -E i )/  -  pert ]t )dt  do the integral….  If you take |c f | 2, you will have the probability that the perturbation will cause a particle in state |  i > to end up in state|  f >  Notice that cf (and thus the probability) is max if one of the denominators is zero.  this is a resonance between the system and the perturbation

7. Resonant Perturbation  Notice that cf (and thus the probability) is max if one of the denominators is zero.  Resonance if   pert = (E f -E i )/   If this is a light-wave perturbing your system, then    pert =energy of a photon  Resonance occurs if the energy of the photon is equal to the energy needed to excite system from initial to final state. Photon is absorbed.  Excitation can occur if  pert is not exactly right, but less likely

8. Resonant Perturbation  But also… resonance if   pert = (E i -E f )/  !  If this is a light-wave perturbing your system, then    pert =energy of a photon  Resonance occurs if the energy of the photon is equal to the energy that will be released when the system DE-excite from initial to final state.  Photon is NOT absorbed.  Photon comes in and stimulates emission of another photon of the same frequency.

9. LaSEr  This is how a laser works  Light interacts with atoms in excited state.  Photons stimulate emission of more photons of the same frequency, creating more light,  without absorbing the original light.  More photons means the light is amplified  LASER = Light Amplification through Stimulated Emission of Radiation

10. PAL